Biostatistics Advance Access originally published online on July 11, 2007
Biostatistics 2008 9(1):172-186; doi:10.1093/biostatistics/kxm023
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An alternative model for bivariate random-effects meta-analysis when the within-study correlations are unknown
Centre for Medical Statistics and Health Evaluation, Faculty of Medicine, University of Liverpool, Shelley's Cottage, Brownlow Street, Liverpool, England L69 3GS richard.riley{at}liv.ac.uk
Centre for Biostatistics and Genetic Epidemiology, Department of Health Sciences, University of Leicester, Second Floor, Adrian Building, University Road, Leicester, England LE1 7RH
* To whom correspondence should be addressed.
Multivariate meta-analysis models can be used to synthesize multiple, correlated endpoints such as overall and disease-free survival. A hierarchical framework for multivariate random-effects meta-analysis includes both within-study and between-study correlation. The within-study correlations are assumed known, but they are usually unavailable, which limits the multivariate approach in practice. In this paper, we consider synthesis of 2 correlated endpoints and propose an alternative model for bivariate random-effects meta-analysis (BRMA). This model maintains the individual weighting of each study in the analysis but includes only one overall correlation parameter, 
, which removes the need to know the within-study correlations. Further, the only data needed to fit the model are those required for a separate univariate random-effects meta-analysis (URMA) of each endpoint, currently the common approach in practice. This makes the alternative model immediately applicable to a wide variety of evidence synthesis situations, including studies of prognosis and surrogate outcomes. We examine the performance of the alternative model through analytic assessment, a realistic simulation study, and application to data sets from the literature. Our results show that, unless 
is very close to 1 or –1, the alternative model produces appropriate pooled estimates with little bias that (i) are very similar to those from a fully hierarchical BRMA model where the within-study correlations are known and (ii) have better statistical properties than those from separate URMAs, especially given missing data. The alternative model is also less prone to estimation at parameter space boundaries than the fully hierarchical model and thus may be preferred even when the within-study correlations are known. It also suitably estimates a function of the pooled estimates and their correlation; however, it only provides an approximate indication of the between-study variation. The alternative model greatly facilitates the utilization of correlation in meta-analysis and should allow an increased application of BRMA in practice.
Keywords: Correlation; Evidence synthesis; Multiple outcomes; Multivariate random-effects meta-analysis; Systematic review
Received October 4, 2006; revised January 31, 2007; revised March 5, 2007; accepted for publication April 25, 2007.